Abstract
AbstractWe compute projective dimension of translated simple modules in the regular block of the BernsteinâGelfandâGelfand category $\mathcal{O}$ in terms of KazhdanâLusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan KĂ„hrström. We also derive some inequalities, in terms of Lusztigâs $\textbf{a}$-function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we prove that Kostantâs problem is equivalent to a homological problem of decomposing translated simple modules in $\mathcal O$. This gives a conjectural answer to Kostantâs problem in terms of the KazhdanâLusztig basis and addresses yet another conjecture by Johan KĂ„hrström.
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